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If you can weigh all the even numbers, you can define all the odd numbers. The question you linked to has the weights as powers of $3: 3^0=1, 3^1=3, 3^2=9, 3^3=27$ and can weigh all numbers up to $40$. So doubling the weights (as you started) $2,6,18,54$ allows you to define all weights up to $81$. More generally, the series of weights $2\cdot 3^i$, for $i=0,1,2,\dots n$ lets you define all weights up to $3^{n+1}$